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16 COMPACT SPACES 59

balls in the following way: from the assumption it follows that the diameter
of E is finite, and by multiplying the distance on E by a constant, we may
assume that <5(E) < 1/2, hence E is a ball B₀ of radius 1. Suppose the B_fchave been defined for 0 < fc < w — 1, and that for these values of k, B_k has a
radius equal to l/2^fc, and there is no finite subfamily of (U_A)_AgL which is a
covering of B_k . Then we consider a finite covering (V_fc)₁^_fc^_m of E by balls
of radius 1/2"; among the balls V_k which have a nonempty intersection with
B,,-!, there is one at least B_n for which no finite subfamily of (U_A) is a covering;
otherwise, as these V_k form a covering of B_n^_l7 there would be a finite sub-
family of (U^) which would be a covering of B_n-ii the induction can thus
proceed indefinitely. Let x_n be the center of B_M; as B_n»₁ and B_M have a com-
mon point, the triangle inequality shows that

Hence, if /?</?< q, we have

This proves that (x_n) is a Cauchy sequence in E, hence converges to a point a.
Let A₀ be an index such that a e U_Ao; there is an a > 0 such that B(a; a) c U_Ao .
From the definition of a, it follows there exists an integer n such that
d(a, x_n) < a/2, and l/2"<a/2. The triangle inequality then shows that
B_n c. B(a; a) c: U_Ao. But this is a contradiction since no finite subfamily
of (UJ is supposed to be a covering of B,, .

(3.16.2) Any precompact metric space is separable.

If E is precompact, for any n there is, by definition, a finite subset A_nof E such that for every x e E₅ d(x, A_n) < 1/n. Let A = \J A_n; A is at most

denumerable, and for each x e E, d(x, A) < d(x, A_B) < l/n for any n, hence
rf(jc, A) = 0, E=A.

(3.16.3) Let E be a metric space. Any two of the following properties Imply
the third:

(a) E is compact.

(b) E is discrete (more precisely, homeomorphic to a discrete space).



	16 COMPACT SPACES 59 balls in the following way: from the assumption it follows that the diameter of E is finite, and by multiplying the distance on E by a constant, we may assume that <5(E) < 1/2, hence E is a ball B₀ of radius 1. Suppose the B_fchave been defined for 0 < fc < w — 1, and that for these values of k, B_k has a radius equal to l/2^fc, and there is no finite subfamily of (U_A)_AgL which is a covering of B_k . Then we consider a finite covering (V_fc)₁^_fc^_m of E by balls of radius 1/2"; among the balls V_k which have a nonempty intersection with B,,-!, there is one at least B_n for which no finite subfamily of (U_A) is a covering; otherwise, as these V_k form a covering of B_n^_l7 there would be a finite sub- family of (U^) which would be a covering of B_n-ii the induction can thus proceed indefinitely. Let x_n be the center of B_M; as B_n»₁ and B_M have a com- mon point, the triangle inequality shows that

	Hence, if /?</?< q, we have

	This proves that (x_n) is a Cauchy sequence in E, hence converges to a point a. Let A₀ be an index such that a e U_Ao; there is an a > 0 such that B(a; a) c U_Ao . From the definition of a, it follows there exists an integer n such that d(a, x_n) < a/2, and l/2"<a/2. The triangle inequality then shows that B_n c. B(a; a) c: U_Ao. But this is a contradiction since no finite subfamily of (UJ is supposed to be a covering of B,, .

	(3.16.2) Any precompact metric space is separable. If E is precompact, for any n there is, by definition, a finite subset A_nof E such that for every x e E₅ d(x, A_n) < 1/n. Let A = \J A_n; A is at most n denumerable, and for each x e E, d(x, A) < d(x, A_B) < l/n for any n, hence rf(jc, A) = 0, E=A.

	(3.16.3) Let E be a metric space. Any two of the following properties Imply the third: (a) E is compact. (b) E is discrete (more precisely, homeomorphic to a discrete space). (c) E is finite.